ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Grigorieva E. V., Kashchenko I. S., Kashchenko S. A. Hypermultistability in laser’s models with large delay. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 3, pp. 3-15. DOI: 10.18500/0869-6632-2011-19-3-3-15

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2011-19-3-3-15

Hypermultistability in laser’s models with large delay

Autors: 
Grigorieva Elena Viktorovna, Belarus State Economic University (BSEU)
Kashchenko I. S., P. G. Demidov Yaroslavl State University
Kashchenko Sergej Aleksandrovich, P. G. Demidov Yaroslavl State University
Abstract: 

We study model of monomode semiconductor laser with optoelectronic feedback, based on balanced equations with delay. We built sets of quasinormal forms in neighboorghood of bifurcation values. The possibility of coexistence of large amount of stable oscillating solutions is shown.

Key words: 
Method of normal forms
large delay
small parameter
local analysis
Reference: 
  1. Yanchuk S, Perlikowski P. Delay and periodicity. Phys. Rev. E. 2009;79(4):046221. DOI: 10.1103/PhysRevE.79.046221.
  2. Loose A, Goswami BK, Wunsche HJ, Henneberger F. Tristability of a semiconductor laser due to time-delayed optical feedback. Phys. Rev. E. 2009;79(3):036211. DOI: 10.1103/PhysRevE.79.036211.
  3. Erneux T, Grasman J. Limit-cycle oscillators subject to a delayed feedback. Phys. Rev. E. 2008;78(2):026209. DOI: 10.1103/PhysRevE.78.026209.
  4. Grigorieva EV, Kaschenko SA, Loiko NA, Samson AM. Nonlinear dynamics in a laser with a negative delayed feedback. Physica D. 1992;59(4):297–319. DOI: 10.1016/0167-2789(92)90072-U.
  5. Grigorieva EV, Kaschenko SА. Regular and chaotic pulsations in laser diode with delayed feedback. Int. J. Bifurcat. Chaos. 1993;3(6):1515–1528. DOI: 10.1142/S0218127493001197.
  6. Wolfrum M, Yanchuk S. Eckhaus instability in systems with large delay. Phys. Rev. Lett. 2006;96(22):220201. DOI: 10.1103/PhysRevLett.96.220201.
  7. Paoli TL, Ripper LE. Frequency stabilization and narrowing of optical pulses from CW GaAs injection lasers. IEEE J. Quan. Electron. 1970;6(6):335–339. DOI: 10.1109/JQE.1970.1076462.
  8. Giacomelli G, Calzavara M, Arecchi FT. Instabilities in a semiconductor laser with delayed optoelectronic feedback. Opt. Commun. 1989;74(1–2):97–101. DOI: 10.1016/0030-4018(89)90498-7.
  9. Arecchi FT, Giacomelli G, Lapucci A, Meucci R. Dynamics of a CO2 laser with delayed feedback: The short-delayed regime. Phys. Rev. A. 1991;43(9):4997–5004. DOI: 10.1103/physreva.43.4997.
  10. Kaschenko SA. Investigation by large parameter methods of a system of nonlinear differential-difference equations simulating the predator-prey problem. Sov. Math. Dokl. 1982;266(4):792–795 (in Russian).
  11. Kaschenko SA. On steady-state regimes of the Hutchinson equation with diffusion. Sov. Math. Dokl. 1987;292(2):327–330 (in Russian).
  12. Butuzov VF, Vasilieva AB. Asymptotic Expansions of Solutions of Singularly Perturbed Equations. Moscow: Nauka; 1973. 272 p. (in Russian).
  13. Grigorieva EV, Haken H, Kaschenko SA. Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback. Opt. Commun. 1999;165(4–6):279–292. DOI: 10.1016/S0030-4018(99)00236-9.
  14. Bestehorn M, Grigorieva EV, Haken H, and Kaschenko SA. Order parameters for class-B lasers with a long time delayed feedback. Physica D. 2000;145(1–2):111–129. DOI: 10.1016/S0167-2789(00)00106-8.
  15. Kaschenko SA. Quasinormal forms for parabolic equations with small diffusion. Sov. Math. Dokl. 1988;37(2):510–513.
  16. Kaschenko SA. Short-wave bifurcations in systems with small diffusion. Sov. Math. Dokl. 1990;40(1):54–58.
  17. Kaschenko SA. Application of the normalization method to the study of the dynamics of differential-difference equations with a small factor at the derivative. Differential Equations. 1989;25(8):1448–1451 (in Russian).
  18. Kaschenko S.A. Normalization in the systems with small diffusion. Int. J. Bifurcat. Chaos. 1996;6(6):1093–1109. DOI: 10.1142/S021812749600059X.
  19. Kashchenko SA. The Ginzburg – Landau equation as a normal form for a second-order difference-differential equation with a large delay. Comput. Math. Math. Phys. 1998;38(3):443–451.
  20. Malinetskii GG, Kurdyumov SP, editors. New in Synergetics. A Look into the Third Millennium. Moscow: Nauka; 2002. 478 p. (in Russian).
  21. Kashchenko IS. Asymptotic analysis of the behavior of solutions to equations with large delay. Dokl. Math. 2008;78(1):570–573. DOI: 10.1134/S1064562408040261.
  22. Kashchenko IS. Local dynamics of equations with large delay. Comput. Math. Math. Phys. 2008;48(12):2172–2181. DOI: 10.1134/S0965542508120075.
  23. Kaschenko IS. The buffer phenomenon in second-order equations with large delay. Automatic Control and Computer Sciences. 2008;15(2):31–35 (in Russian).  
Received: 
15.02.2011
Accepted: 
15.02.2011
Published: 
29.07.2011
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